文档介绍：Investment Tools – Probability
SASF CFA Quant. Review
1
Probability
A random variable is a quantity whose e is uncertain.
Two defining properties of Probability.
Probability of any event E is a number between 0 and 1, p(E).
Sum of the probabilities of any list of mutually exclusive and exhaustive events equals 1.
Mutually exclusive = one and only one event can occur at any time.
Exhaustive = one of the events must occur, jointly cover all possible es.
Empirical probability - probability of an event occurring is estimated from data, usually in the form of a relative frequency.
A priori probability - probability of an event is deduced by reasoning about the structure of the problem itself.
Subjective probability - probability of an event is based on a personal assessment without reference to any particular data.
2
Visualizing Sample Space
1. Listing
S = {Head, Tail}
2. Contingency Table
3. Decision Tree Diagram
3
2
nd
Coin
1
st
Coin
Head
Tail
Total
Head
HH
HT
HH, HT
Tail
TH
TT
TH, TT
Total
HH,
TH
HT,
TT
S
Contingency Table
Experiment: Toss 2 Coins. Note Faces.
S = {HH, HT, TH, TT}
Sample Space
e (Count, Total % Shown Usually)
Simple Event (Head on 1st Coin)
4
Tree Diagram
e
S = {HH, HT, TH, TT}
Sample Space
Experiment: Toss 2 Coins. Note Faces.
T
H
T
H
T
HH
HT
TH
TT
H
5
Second Die
First
Die
Probabilities for Two Dice
6
pound Events
1. Intersection
es in Both Events A and B
‘AND’ Statement
Symbol (., A B)
2. Union
es in Either Events A or B or Both
‘OR’ Statement
Symbol (., A B)
7
Answer: Count binations that yield 7 and add the individual probabilities
to find the probability of the event, A, “Rolling a seven”
P(A) = P(A B1) + P(A B2) + P(AB3) +…+ P(A B6) = 6/36 = 1/6
Second Die
First
Die
Joint Probabilities for Two Dice
What is the Probability of
throwing a Seven
with the two Dice?
8
Addition Rule
1. Used to pound Probabilities for Union of Events:
P(A OR B) = P(A B)